Transition from Poisson bracket into Canonical Commutation Relations

[exponent137]

In book
http://www.phy.uct.ac.za/people/horowitz/Teaching/lecturenotes.pdf
in section 2 it is described transition from Poisson bracket into Canonical Commutation Relations.

But it is written

The experimentally observed phenomenon of incompatible measurements suggests that position and momentum cannot be number-valued functions in QM.

What means "number-valued functions"?
I understand QM, but I do not understand this word, what he want to tell with it.

 

[tom.stoer]

It means that x and p as position and momentum acting on wave functions ψ cannot be both numbers; usually one choses a representation where e.g. x is a number and p is an operator like -i∂_x acting on a wave function ψ(x) in position space.

 

[Jano L.]

It means that the experiments considered are not capable of determining both the position and the momentum of the particle accurately at the same time.

In quantum theory, this incapability is mathematically formulated in terms of the wave function. There are no positions and momenta as functions of time in the theory. The wave function can be used to determine the probability that such and such coordinate / momentum will be measured in particular experiment.

For example, if the wave function is localized around some coordinate x_0, the calculated statistical scatter of the position measurements is small. However, the statistical scatter of the measured momenta on the same ensemble of particles is calculated to be great.

However, this character of quantum theory do not necessarily imply that the particle cannot be ascribed coordinate and momentum at the same time. It is only incapable to determine them and goes on without them. There is a version of quantum theory where particles do have coordinates and momenta simultaneously, but their roles are hidden. It is called Bohmian mechanics.